It appears that the problem of showing that every positive integer is the class number of some quadratic number field seems rather intractable, despite the wide belief that it is true.

So I ask an easier question: For each positive number $x$, let $\rho(x)$ denote the number of positive integers $n \leq x$ such that $n$ cannot be realized as the class number of a quadratic number field. Then is it possible to show that $\rho(x)/x$ tends to 0? Better yet, is it possible to show that $\rho(x) \ll x^\delta$ for some $0 < \delta < 1$?

If $\rho(x) \ll x^\delta$ isn't a 'natural' upperbound, then what would be? Can one expect a better or worse upper bound? Can a bound of the form $\rho(x) \ll x/(\log x)^A$ for some $A > 0$ be likely?

Seems very doubtful to me that you could prove any of this using current techniques. Of course, I would be very happy to be demonstrated wrong!
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Frank ThorneMar 28 '12 at 18:02

Is it plausible to use the theory of Hilbert class fields to construct a number field with a given class number?
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Stanley Yao XiaoMar 28 '12 at 23:11

Observe that if you replace the word "number" by "group", the answer is unknown (see Washington's book, second edition, page 27, last two lines). But if you jus want \emph{a bound}, you can certainly do, as in Corollary 3.9 (same book, same page).
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Filippo Alberto EdoardoApr 6 '12 at 9:59